The team's victory marked the first time Amorim has earned two consecutive home Premier League wins since taking over.
Chelsea goalkeeper Robert Sanchez was given a red card just five minutes into the game after a foul on Bryan Mbeumo.
Midfielder Bruno Fernandes scored the opening goal, followed by Casemiro's strike just before he received a second yellow card in the first half's added time.
Trevoh Chalobah scored in the final 10 minutes to make the game more intense, but Manchester United held on to secure a much-needed win in the rain.
Amorim, who has built a strong connection with the fans despite previous poor results, stayed late to thank the supporters for their loyalty.
The team will aim to build on this win as they travel to Brentford for their next Premier League match on September 27.